LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine strsen ( character JOB, character COMPQ, logical, dimension( * ) SELECT, integer N, real, dimension( ldt, * ) T, integer LDT, real, dimension( ldq, * ) Q, integer LDQ, real, dimension( * ) WR, real, dimension( * ) WI, integer M, real S, real SEP, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO )

STRSEN

Purpose:
``` STRSEN reorders the real Schur factorization of a real matrix
A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
the leading diagonal blocks of the upper quasi-triangular matrix T,
and the leading columns of Q form an orthonormal basis of the
corresponding right invariant subspace.

Optionally the routine computes the reciprocal condition numbers of
the cluster of eigenvalues and/or the invariant subspace.

T must be in Schur canonical form (as returned by SHSEQR), that is,
block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
2-by-2 diagonal block has its diagonal elements equal and its
off-diagonal elements of opposite sign.```
Parameters
 [in] JOB ``` JOB is CHARACTER*1 Specifies whether condition numbers are required for the cluster of eigenvalues (S) or the invariant subspace (SEP): = 'N': none; = 'E': for eigenvalues only (S); = 'V': for invariant subspace only (SEP); = 'B': for both eigenvalues and invariant subspace (S and SEP).``` [in] COMPQ ``` COMPQ is CHARACTER*1 = 'V': update the matrix Q of Schur vectors; = 'N': do not update Q.``` [in] SELECT ``` SELECT is LOGICAL array, dimension (N) SELECT specifies the eigenvalues in the selected cluster. To select a real eigenvalue w(j), SELECT(j) must be set to .TRUE.. To select a complex conjugate pair of eigenvalues w(j) and w(j+1), corresponding to a 2-by-2 diagonal block, either SELECT(j) or SELECT(j+1) or both must be set to .TRUE.; a complex conjugate pair of eigenvalues must be either both included in the cluster or both excluded.``` [in] N ``` N is INTEGER The order of the matrix T. N >= 0.``` [in,out] T ``` T is REAL array, dimension (LDT,N) On entry, the upper quasi-triangular matrix T, in Schur canonical form. On exit, T is overwritten by the reordered matrix T, again in Schur canonical form, with the selected eigenvalues in the leading diagonal blocks.``` [in] LDT ``` LDT is INTEGER The leading dimension of the array T. LDT >= max(1,N).``` [in,out] Q ``` Q is REAL array, dimension (LDQ,N) On entry, if COMPQ = 'V', the matrix Q of Schur vectors. On exit, if COMPQ = 'V', Q has been postmultiplied by the orthogonal transformation matrix which reorders T; the leading M columns of Q form an orthonormal basis for the specified invariant subspace. If COMPQ = 'N', Q is not referenced.``` [in] LDQ ``` LDQ is INTEGER The leading dimension of the array Q. LDQ >= 1; and if COMPQ = 'V', LDQ >= N.``` [out] WR ` WR is REAL array, dimension (N)` [out] WI ``` WI is REAL array, dimension (N) The real and imaginary parts, respectively, of the reordered eigenvalues of T. The eigenvalues are stored in the same order as on the diagonal of T, with WR(i) = T(i,i) and, if T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and WI(i+1) = -WI(i). Note that if a complex eigenvalue is sufficiently ill-conditioned, then its value may differ significantly from its value before reordering.``` [out] M ``` M is INTEGER The dimension of the specified invariant subspace. 0 < = M <= N.``` [out] S ``` S is REAL If JOB = 'E' or 'B', S is a lower bound on the reciprocal condition number for the selected cluster of eigenvalues. S cannot underestimate the true reciprocal condition number by more than a factor of sqrt(N). If M = 0 or N, S = 1. If JOB = 'N' or 'V', S is not referenced.``` [out] SEP ``` SEP is REAL If JOB = 'V' or 'B', SEP is the estimated reciprocal condition number of the specified invariant subspace. If M = 0 or N, SEP = norm(T). If JOB = 'N' or 'E', SEP is not referenced.``` [out] WORK ``` WORK is REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.``` [in] LWORK ``` LWORK is INTEGER The dimension of the array WORK. If JOB = 'N', LWORK >= max(1,N); if JOB = 'E', LWORK >= max(1,M*(N-M)); if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)). If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.``` [out] IWORK ``` IWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.``` [in] LIWORK ``` LIWORK is INTEGER The dimension of the array IWORK. If JOB = 'N' or 'E', LIWORK >= 1; if JOB = 'V' or 'B', LIWORK >= max(1,M*(N-M)). If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to LIWORK is issued by XERBLA.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value = 1: reordering of T failed because some eigenvalues are too close to separate (the problem is very ill-conditioned); T may have been partially reordered, and WR and WI contain the eigenvalues in the same order as in T; S and SEP (if requested) are set to zero.```
Date
April 2012
Further Details:
```  STRSEN first collects the selected eigenvalues by computing an
orthogonal transformation Z to move them to the top left corner of T.
In other words, the selected eigenvalues are the eigenvalues of T11
in:

Z**T * T * Z = ( T11 T12 ) n1
(  0  T22 ) n2
n1  n2

where N = n1+n2 and Z**T means the transpose of Z. The first n1 columns
of Z span the specified invariant subspace of T.

If T has been obtained from the real Schur factorization of a matrix
A = Q*T*Q**T, then the reordered real Schur factorization of A is given
by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, and the first n1 columns of Q*Z span
the corresponding invariant subspace of A.

The reciprocal condition number of the average of the eigenvalues of
T11 may be returned in S. S lies between 0 (very badly conditioned)
and 1 (very well conditioned). It is computed as follows. First we
compute R so that

P = ( I  R ) n1
( 0  0 ) n2
n1 n2

is the projector on the invariant subspace associated with T11.
R is the solution of the Sylvester equation:

T11*R - R*T22 = T12.

Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
the two-norm of M. Then S is computed as the lower bound

(1 + F-norm(R)**2)**(-1/2)

on the reciprocal of 2-norm(P), the true reciprocal condition number.
S cannot underestimate 1 / 2-norm(P) by more than a factor of
sqrt(N).

An approximate error bound for the computed average of the
eigenvalues of T11 is

EPS * norm(T) / S

where EPS is the machine precision.

The reciprocal condition number of the right invariant subspace
spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
SEP is defined as the separation of T11 and T22:

sep( T11, T22 ) = sigma-min( C )

where sigma-min(C) is the smallest singular value of the
n1*n2-by-n1*n2 matrix

C  = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )

I(m) is an m by m identity matrix, and kprod denotes the Kronecker
product. We estimate sigma-min(C) by the reciprocal of an estimate of
the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).

When SEP is small, small changes in T can cause large changes in
the invariant subspace. An approximate bound on the maximum angular
error in the computed right invariant subspace is

EPS * norm(T) / SEP```

Definition at line 316 of file strsen.f.

316 *
317 * -- LAPACK computational routine (version 3.4.1) --
318 * -- LAPACK is a software package provided by Univ. of Tennessee, --
319 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
320 * April 2012
321 *
322 * .. Scalar Arguments ..
323  CHARACTER compq, job
324  INTEGER info, ldq, ldt, liwork, lwork, m, n
325  REAL s, sep
326 * ..
327 * .. Array Arguments ..
328  LOGICAL select( * )
329  INTEGER iwork( * )
330  REAL q( ldq, * ), t( ldt, * ), wi( * ), work( * ),
331  \$ wr( * )
332 * ..
333 *
334 * =====================================================================
335 *
336 * .. Parameters ..
337  REAL zero, one
338  parameter ( zero = 0.0e+0, one = 1.0e+0 )
339 * ..
340 * .. Local Scalars ..
341  LOGICAL lquery, pair, swap, wantbh, wantq, wants,
342  \$ wantsp
343  INTEGER ierr, k, kase, kk, ks, liwmin, lwmin, n1, n2,
344  \$ nn
345  REAL est, rnorm, scale
346 * ..
347 * .. Local Arrays ..
348  INTEGER isave( 3 )
349 * ..
350 * .. External Functions ..
351  LOGICAL lsame
352  REAL slange
353  EXTERNAL lsame, slange
354 * ..
355 * .. External Subroutines ..
356  EXTERNAL slacn2, slacpy, strexc, strsyl, xerbla
357 * ..
358 * .. Intrinsic Functions ..
359  INTRINSIC abs, max, sqrt
360 * ..
361 * .. Executable Statements ..
362 *
363 * Decode and test the input parameters
364 *
365  wantbh = lsame( job, 'B' )
366  wants = lsame( job, 'E' ) .OR. wantbh
367  wantsp = lsame( job, 'V' ) .OR. wantbh
368  wantq = lsame( compq, 'V' )
369 *
370  info = 0
371  lquery = ( lwork.EQ.-1 )
372  IF( .NOT.lsame( job, 'N' ) .AND. .NOT.wants .AND. .NOT.wantsp )
373  \$ THEN
374  info = -1
375  ELSE IF( .NOT.lsame( compq, 'N' ) .AND. .NOT.wantq ) THEN
376  info = -2
377  ELSE IF( n.LT.0 ) THEN
378  info = -4
379  ELSE IF( ldt.LT.max( 1, n ) ) THEN
380  info = -6
381  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
382  info = -8
383  ELSE
384 *
385 * Set M to the dimension of the specified invariant subspace,
386 * and test LWORK and LIWORK.
387 *
388  m = 0
389  pair = .false.
390  DO 10 k = 1, n
391  IF( pair ) THEN
392  pair = .false.
393  ELSE
394  IF( k.LT.n ) THEN
395  IF( t( k+1, k ).EQ.zero ) THEN
396  IF( SELECT( k ) )
397  \$ m = m + 1
398  ELSE
399  pair = .true.
400  IF( SELECT( k ) .OR. SELECT( k+1 ) )
401  \$ m = m + 2
402  END IF
403  ELSE
404  IF( SELECT( n ) )
405  \$ m = m + 1
406  END IF
407  END IF
408  10 CONTINUE
409 *
410  n1 = m
411  n2 = n - m
412  nn = n1*n2
413 *
414  IF( wantsp ) THEN
415  lwmin = max( 1, 2*nn )
416  liwmin = max( 1, nn )
417  ELSE IF( lsame( job, 'N' ) ) THEN
418  lwmin = max( 1, n )
419  liwmin = 1
420  ELSE IF( lsame( job, 'E' ) ) THEN
421  lwmin = max( 1, nn )
422  liwmin = 1
423  END IF
424 *
425  IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
426  info = -15
427  ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
428  info = -17
429  END IF
430  END IF
431 *
432  IF( info.EQ.0 ) THEN
433  work( 1 ) = lwmin
434  iwork( 1 ) = liwmin
435  END IF
436 *
437  IF( info.NE.0 ) THEN
438  CALL xerbla( 'STRSEN', -info )
439  RETURN
440  ELSE IF( lquery ) THEN
441  RETURN
442  END IF
443 *
444 * Quick return if possible.
445 *
446  IF( m.EQ.n .OR. m.EQ.0 ) THEN
447  IF( wants )
448  \$ s = one
449  IF( wantsp )
450  \$ sep = slange( '1', n, n, t, ldt, work )
451  GO TO 40
452  END IF
453 *
454 * Collect the selected blocks at the top-left corner of T.
455 *
456  ks = 0
457  pair = .false.
458  DO 20 k = 1, n
459  IF( pair ) THEN
460  pair = .false.
461  ELSE
462  swap = SELECT( k )
463  IF( k.LT.n ) THEN
464  IF( t( k+1, k ).NE.zero ) THEN
465  pair = .true.
466  swap = swap .OR. SELECT( k+1 )
467  END IF
468  END IF
469  IF( swap ) THEN
470  ks = ks + 1
471 *
472 * Swap the K-th block to position KS.
473 *
474  ierr = 0
475  kk = k
476  IF( k.NE.ks )
477  \$ CALL strexc( compq, n, t, ldt, q, ldq, kk, ks, work,
478  \$ ierr )
479  IF( ierr.EQ.1 .OR. ierr.EQ.2 ) THEN
480 *
481 * Blocks too close to swap: exit.
482 *
483  info = 1
484  IF( wants )
485  \$ s = zero
486  IF( wantsp )
487  \$ sep = zero
488  GO TO 40
489  END IF
490  IF( pair )
491  \$ ks = ks + 1
492  END IF
493  END IF
494  20 CONTINUE
495 *
496  IF( wants ) THEN
497 *
498 * Solve Sylvester equation for R:
499 *
500 * T11*R - R*T22 = scale*T12
501 *
502  CALL slacpy( 'F', n1, n2, t( 1, n1+1 ), ldt, work, n1 )
503  CALL strsyl( 'N', 'N', -1, n1, n2, t, ldt, t( n1+1, n1+1 ),
504  \$ ldt, work, n1, scale, ierr )
505 *
506 * Estimate the reciprocal of the condition number of the cluster
507 * of eigenvalues.
508 *
509  rnorm = slange( 'F', n1, n2, work, n1, work )
510  IF( rnorm.EQ.zero ) THEN
511  s = one
512  ELSE
513  s = scale / ( sqrt( scale*scale / rnorm+rnorm )*
514  \$ sqrt( rnorm ) )
515  END IF
516  END IF
517 *
518  IF( wantsp ) THEN
519 *
520 * Estimate sep(T11,T22).
521 *
522  est = zero
523  kase = 0
524  30 CONTINUE
525  CALL slacn2( nn, work( nn+1 ), work, iwork, est, kase, isave )
526  IF( kase.NE.0 ) THEN
527  IF( kase.EQ.1 ) THEN
528 *
529 * Solve T11*R - R*T22 = scale*X.
530 *
531  CALL strsyl( 'N', 'N', -1, n1, n2, t, ldt,
532  \$ t( n1+1, n1+1 ), ldt, work, n1, scale,
533  \$ ierr )
534  ELSE
535 *
536 * Solve T11**T*R - R*T22**T = scale*X.
537 *
538  CALL strsyl( 'T', 'T', -1, n1, n2, t, ldt,
539  \$ t( n1+1, n1+1 ), ldt, work, n1, scale,
540  \$ ierr )
541  END IF
542  GO TO 30
543  END IF
544 *
545  sep = scale / est
546  END IF
547 *
548  40 CONTINUE
549 *
550 * Store the output eigenvalues in WR and WI.
551 *
552  DO 50 k = 1, n
553  wr( k ) = t( k, k )
554  wi( k ) = zero
555  50 CONTINUE
556  DO 60 k = 1, n - 1
557  IF( t( k+1, k ).NE.zero ) THEN
558  wi( k ) = sqrt( abs( t( k, k+1 ) ) )*
559  \$ sqrt( abs( t( k+1, k ) ) )
560  wi( k+1 ) = -wi( k )
561  END IF
562  60 CONTINUE
563 *
564  work( 1 ) = lwmin
565  iwork( 1 ) = liwmin
566 *
567  RETURN
568 *
569 * End of STRSEN
570 *
subroutine strsyl(TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C, LDC, SCALE, INFO)
STRSYL
Definition: strsyl.f:166
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:105
subroutine strexc(COMPQ, N, T, LDT, Q, LDQ, IFST, ILST, WORK, INFO)
STREXC
Definition: strexc.f:148
real function slange(NORM, M, N, A, LDA, WORK)
SLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: slange.f:116
subroutine slacn2(N, V, X, ISGN, EST, KASE, ISAVE)
SLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition: slacn2.f:138
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55

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